FINDING: Renormalization group reveals scale invariance and power-law criticality in Ising model and complex systems, linking microscopic fluctuations to macroscopic universality. MATH: - Critical exponents (e. g. , ν, β, γ) from RG flow equations: \ (dgd = (g) \), fixed points \ ( (g^*) = 0 \). - Power-law correlations: \ (sᵢ sⱼ |i-j|^- (d-2+) \) at criticality. - Scaling relations: \ (+ 2 + = 2 \), \ (= (2-) \). - Partition function: \ (Z = \ₒ\ e^- H \), with \ (H = -J ₈, ₉ sᵢ sⱼ \). - Cantor set dimension: \ (D = 2 / 3 0. 6309 \). CONNECTION: - Scale invariance at critical point mirrors geometric self-similarity: ratios 0. 618 (golden mean conjugate) appear in critical exponent relations (e. g. , \ (0. 63 \) for 2D Ising, close to \ (2 / 3 \) ). - Lattice structure (square/hexagonal) relates to crystallographic s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Wed,) studied this question.